The Step Function and the Signum Function

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On this page, we'll look at the Fourier Transform for some useful functions, the step function, u(t), and the signum function, sgn(t). The function u(t) is defined mathematically in equation [1], and the signum function is defined in equation [2]:

unit step function
[Equation 1]

signum function
[Equation 2]

The signum function is also known as the "sign" function, because if t is positive, the signum function is +1; if t is negative, the signum function is -1. The unit step function "steps" up from 0 to 1 at t=0.

The unit step (on the left) and the signum function multiplied by 0.5 are plotted in Figure 1:

the unit step functionthe signum function

Figure 1. The Step Function u(t) [left] and 0.5*sgn(t) [right].

The reason we plot one half of the signum function in Figure 1, is that we can see that the unit step function and the signum function are the same, just offset by 0.5 from each other in amplitude.

For the functions in Figure 1, note that they have the same derivative, which is the dirac-delta impulse:

derivative of step function is the impulse       [3]

To obtain the Fourier Transform for the signum function, we will use the results of equation [3], the integration property of Fourier Transforms, and the the fourier transform of the impulse. The integral of the signum function is zero:

integral of signum function       [5]

The Fourier Transform of the signum function can be easily found:

      [6]

The average value of the unit step function is not zero, so the integration property is slightly more difficult to apply. Note that the following equation is true:

average value of unit step       [7]

Hence, the d.c. term is c=0.5, and we can apply the integration property of the Fourier Transform, which gives us the end result:

fourier transform of step function       [8]

The integration property makes the Fourier Transforms of these functions simple to obtain, because we know the Fourier Transform of their derivatives.


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